Beta distribution


Format: Beta(a, b)

 

The Beta distribution in its standard form ranges from zero to one, and takes a wide range of shapes. Examples of the Beta distribution are shown below:

 

Uses

The Beta distribution has two main uses:

  • As the description of uncertainty or random variation of a probability, fraction or prevalence;

  • As a useful distribution one can rescale and shift to create distributions with a wide range of  shapes and  over any finite range. As such, it is sometimes used to model expert opinion, for example in the form of the PERT distribution.

Example 1: A bank may have data on the number of creditors of a certain type that have defaulted (s) out of the total number of creditors of this type. Then the probability that the next creditor of the same type will default can be estimated as Beta(s+1, n-s+1);

Example 2: A random survey of 100 car owners over 65 years of age reveals that 57 considered a newly proposed insurance policy to be more attractive than their current policy. One can estimate that the fraction of drivers in this age group who would have the same opinion as Beta(57+1, 100-57+1) = Beta(58,44).

The Beta distribution is the conjugate prior (meaning it has the same functional form, therefore also often called "convenience prior")  to the Binomial likelihood function in Bayesian inference and, as such, is often used to describe the uncertainty about a binomial probability, given a number of trials n have been made with a number of recorded successes s. In these situations, a  is set to the value (s + x) and b is set to (n - s + y), where Beta(x, y) is the prior.

Comments

A Beta(1, 1) = Uniform(0, 1) is usually used as a non-informative prior, though a Beta(½,½) and a Beta(0,0) are also sometimes used.

The Beta distribution has also been used for a wide variety of other applications because it can take a very diverse set of shapes, as illustrated in the graphs above. The Beta distribution can be rescaled to model a variable that runs from a to b by using the formula:

x = a + Beta (a ,b) * (b - a)

This is the four-parameter version of the Beta distribution. It is implemented in ModelRisk directly as the Beta4 distribution.

A version of this four-parameter Beta distribution is called a PERT distribution. It makes the assumption that the mean = (minimum + 4*most likely = maximum) / 6. This extra equation allows the four parameters to be determined from three input values: the minimum, most likely and maximum, which makes it ideal for modeling expert opinion of a variable's uncertainty.

David Vose developed a second version of the PERT distribution called, for want of a better phrase, the Modified PERT distribution. It allows the user to vary the degree of peakedness of the distribution within the constraints of the minimum, most likely and maximum values of the standard PERT distribution.

An alternative to the PERT is the Beta Subjective distribution in which one defines the mean as well as the min, mode and max.

Beta(a, b) = 1 - Beta(b, a): a property that is readily apparent in the context of modeling uncertainty about a binomial probability. This identity is sometimes useful to keep equations neater.

A Beta(z,1) distribution is sometimes called a Power-function distribution with parameter z.

ModelRisk functions added to Microsoft Excel for the Beta distribution

VoseBetaSubj generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseBetaSubjObject constructs a distribution object for this distribution.

VoseBetaSubjProb returns the probability density or cumulative distribution function for this distribution.

VoseBetaSubjProb10 returns the log10 of the probability density or cumulative distribution function. 

VoseBetaSubjFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VoseBetaSubjFitObject constructs a distribution object of this distribution fitted to data.

VoseBetaSubjFitP returns the parameters of this distribution fitted to data.

 

Beta distribution equations

 

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